Sum rules and missing spectral weight in magnetic neutron scattering in the cuprates

We present estimates in the Hubbard and Heisenberg models for the spectral weight in magnetic neutron-scattering experiments on the cuprates. With the aid of spin-wave theory and the time-dependent Gutzwiller approximation, we discuss how the spectral weight is distributed among the different channels and between high and low energies. In addition to the well known total moment sum rule, we discuss sum rules for each component of the dynamical structure factor tensor which are peculiar for spin-$\frac{1}{2}$ systems. The various factors that reduce the spectral weight at the relevant energies are singled out and analyzed, such as shielding factors, weight at electronic energies, multimagnon process, etc. Although about 10–15 % of the naively expected weight is detected in experiments, after consideration of these factors the missing weight is within the experimental uncertainties. A large fraction of the spectral weight is hard to detect with present experimental conditions.

Figure 1

Two-magnon scattering intensity [Eq. (20)] as a function of energy and wave vector along symmetry directions in the Brillouin zone (solid lines in Fig. 2 inset). Density of scattered points represents intensity. The lower boundary traces the one-magnon dispersion relation $\hslash {\omega }_{\mathit{q}}$ and corresponds to events where one of the two magnons has zero energy. The upper bound, $4Z_{C}J$, is reached when both magnons are on the antiferromagnetic zone boundary contour (dashed square in Fig. 2 inset) where they have maximum energy.

Figure 2

Energy integrated spectral weight in the two-magnon continuum (dashed line), $S_{2M}^{zz}\left(\mathit{q}\right)=\int d\left(\hslash \omega \right)S_{2M}^{zz}(\mathit{q},\omega )$, compared with the weight in the one-magnon peak (solid line), from Eq. (22). We have used $Z_d=0.57$ and $Z_{2M}=0.67$.

Figure 3

Momentum integrated spectral function of one- [Eq. (22)] and two-magnon excitations [Eq. (20)]. We have used $Z_d=0.57$ and $Z_{2M}=0.67$. Integrated areas are $(1+2\Delta S)Z_d\mathrm{S}$ and $Z_{2M}\Delta S(1+\Delta S)$, respectively.

Figure 4

(Color online) Single occupation probability (shielding factor) for a half-filled Hubbard model in different approximations. The exact results in a $4\times 4$ site cluster are computed with the double occupancies given in Ref. 27. The apparent lack of extrapolation of the exact results to the noninteracting limit at $U=0$ is due to a finite-size effect.

Figure 5

(Color online) Dispersion relation of the low-energy transverse excitations. We show the experimental result for ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$ after Ref. 30 and the $\mathrm{GA}+\mathrm{RPA}$ result for $U∕t=8$ and $t=335\mathrm{meV}$. The inset shows that for $U∕t=10$ the fit is noticeably worse.

Figure 6

(Color online) $S^{xx}\left(\omega \right)$ evaluated for the Hubbard model within $\mathrm{GA}+\mathrm{RPA}$ for $U∕t=8$ (solid line) and $U∕t=10$ (dashed line) with $Z_d^U=1$ for an $(80\times 80)$-site system. The inset shows the high-energy contribution of $S^{xx}\left(\omega \right)$ at energies $\omega \sim U$.

Figure 7

(Color online) Intensity in the spin-wave excitations as a function of momentum for one transverse spin channel. We show the experimental result for ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$ after Ref. 30 and the $\mathrm{GA}+\mathrm{RPA}$ result for $U∕t=8$ and $t=335\mathrm{meV}$. The $\mathrm{GA}+\mathrm{RPA}$ intensity has been renormalized by a factor $Z_d^{U=8t}=0.65$ and coincides with the spin-wave result [shown in Fig. 3(b) of Ref. 30] renormalized by $Z_d^{\exp }=0.51$ in Eq. (22).

Figure 8

(Color online) Inelastic part of $S^{zz}\left(\omega \right)$ in the $\mathrm{GA}+\mathrm{RPA}$ for different values of $U∕t$ in an $(8\times 8)$-site system.

Figure 9

(Color online) Momentum integrated longitudinal spectral function for the Hubbard model with $U=8t$, $t=335\mathrm{meV}$, and $Z_{c}J=153\mathrm{meV}$. We show the contribution of two-magnon excitations computed in SWT [Eq. (23)] with an effective shielding factor of 0.74 (Table ) and $Z_{2M}=0.67$. We also show the contribution due to scattering across the Hubbard bands computed in $\mathrm{GA}+\mathrm{RPA}$.

Figure 10

(Color online) Zeroth moment and shielding factor as a function of doping. We show the shielding factor for $U∕t=8$ and $t^{\prime }∕t=-0.2$ for the SDW at $x=0$ and bond-centered metallic stripe solutions as a function of doping (solid line) and for $U∕t=\infty$ (dotted line). For the zeroth moment, MBZ means that the integration was restricted to the magnetic Brillouin zone. ${\mathrm{La}}_2\mathrm{Cu}{\mathrm{O}}_4$ (Ref. 30) and CFTD (Ref. 40) label the moments estimated in insulators (Sec. 3E). Their value coincide so CFTD was slightly shifted in doping for clarity. ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x\mathrm{Cu}{\mathrm{O}}_4$ (Ref. 1) labels the experimental value of $M_0$ in the magnetic Brillouin zone, corrected by the polarization factor. In the last case, the error bars indicate the two extreme possibilities for the polarization factor $2∕3$ and $1∕2$ (cf. Sec. 2). The relative true error should be larger than at zero doping, where the error bars have the usual sense. Finally, we show the zeroth moment of the $\mathrm{GA}+\mathrm{RPA}$ transverse spectra at $x=1∕8$ in the whole Brillouin zone (+) and in the MBZ ×. The two lower × include a $Z_d^{U=8t}=0.65$ renormalization factor and for the lower one the energy integration was restricted to energies smaller than $0.23\mathrm{eV}$.

Figure 11

(Color online) Longitudinal and transverse components of the spin autocorrelation function for $x=1∕8$ in a $16\times 4$ site cluster for $U∕t=8$ and $t^{\prime }∕t=-0.2$ and $t=353.7\mathrm{meV}$. Only the inelastic part is shown and we neglect the intensity renormalization $(Z=1)$. The ground state consists of 4 SC stripe running along the short dimension.

Figure 12

(Color online) The upper panel shows the theoretical and the experimental result for $S^{\mathrm{eff}}\left(\omega \right)$ using $U∕t=8$, $t^{\prime }∕t=-0.2$, $t=353.7\mathrm{eV}$, and ${\phantom{⟨}(1-{\widehat{q}}_x^2)⟩}_{\mathrm{dom}}={\phantom{⟨}(1-{\widehat{q}}_y^2)⟩}_{\mathrm{dom}}=1∕2$, in a $(96\times 96)$-site lattice, together with the experimental data for ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x\mathrm{Cu}{\mathrm{O}}_4$ $x=0.125$ (Ref. 1). The lower panel shows the partial sum rule weight. $M_0^{\mathrm{eff}}\left(\omega \right)\equiv (1∕N){\sum }_{\mathit{q}}{\int }_{-0}^{\omega }d\left(\hslash {\omega }^{\prime }\right)S^{\mathrm{eff}}(\mathit{q},{\omega }^{\prime })$.

Figure 13

(Color online) Experimental results for $S^{\mathrm{eff}}\left(\omega \right)$ for $x=0$ (Ref. 45), ${\mathrm{La}}_{2-x}{\mathrm{Ba}}_x\mathrm{Cu}{\mathrm{O}}_4$ $x=0.125$ (Ref. 1), and ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ with $x=0.14$ (Ref. 45) and $x=0.16$ (Ref. 46). We used the following conversions from the quantity reported in the quoted reference: $S^{\mathrm{eff}}=S$ (Ref. 1) $=(2∕\pi )\chi$ (Ref. 45) $=(2∕\pi )\chi$ (Ref. 46).